Zeros of the i.i.d. Gaussian Laurent series on an annulus: weighted Szegő kernels and permanental-determinantal point processes

Abstract

On an annulus ${\mathbb{A}}_q:=\{z\in \mathbb{C}:q<|z|<1\}$ with a fixed $q\in (0,1)$, we study a Gaussian analytic function (GAF) and its zero set which defines a point process on ${\mathbb{A}}_q$ called the zero point process of the GAF. The GAF is defined by the i.i.d.~Gaussian Laurent series such that the covariance kernel parameterized by $r>0$ is identified with the weighted Szegő kernel of ${\mathbb{A}}_q$ with the weight parameter $r$ studied by Mccullough and Shen. The GAF and the zero point process are rotationally invariant and have a symmetry associated with the $q$-inversion of coordinate $z\leftrightarrow q/z$ and the parameter change $r\leftrightarrow q^2/r$. When $r=q$ they are invariant under conformal transformations which preserve ${\mathbb{A}}_q$. Conditioning the GAF by adding zeros, new GAFs are induced such that the covariance kernels are also given by the weighted Szegő kernel of Mccullough and Shen but the weight parameter $r$ is changed depending on the added zeros. We also prove that the zero point process of the GAF provides a permanental-determinantal point process (PDPP) in which each correlation function is expressed by a permanent multiplied by a determinant. Dependence on $r$ of the unfolded 2-correlation function of the PDPP is studied. If we take the limit $q\to 0$, a simpler but still non-trivial PDPP is obtained on the unit disk $\mathbb{D}$. We observe that the limit PDPP indexed by $r\in (0,\infty )$ can be regarded as an interpolation between the determinantal point process (DPP) on $\mathbb{D}$ studied by Peres and Virág ($r\to 0$) and that DPP of Peres and Virág with a deterministic zero added at the origin ($r\to \infty$).

arXiv commentv3: LaTeX 58 pages, 3 figures

References in corpus (20)

• Determinantal random point fields
• Universality and scaling of correlations between zeros on complex manifolds
• Quantum Chaotic Dynamics and Random Polynomials
• Summation and transformation formulas for elliptic hypergeometric series
• Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process
• The O(n) model on the annulus
• The Gaussian free field and SLE(4) on doubly connected domains
• Random Analytic Chaotic Eigenstates
• Elliptic determinant evaluations and the Macdonald identities for affine root systems
• On the equivalence of two fundamental theta identities
• Seeing zeros of random polynomials: quantized vortices in the ideal Bose gas
• Correlation functions for zeros of a Gaussian power series and Pfaffians
• Two-Dimensional Elliptic Determinantal Point Processes and Related Systems
• Liouville quantum gravity on the annulus
• Nonlinear Random Matrix Statistics, symmetric functions and hyperdeterminants
• Particles in a magnetic field and plasma analogies: doubly periodic boundary conditions
• SLE on doubly-connected domains and the winding of loop-erased random walks
• Partial Isometries, Duality, and Determinantal Point Processes
• Scaling limit for determinantal point processes on spheres
• Patterson-Sullivan measures for point processes and the reconstruction of harmonic functions

Cited by in corpus (1)

• Polyanalytic Reproducing Kernels on the Quantized Annulus